Solving ill-posed inverse problems using iterative deep neural networks

We propose a partially learned approach for the solution of ill posed inverse problems with not necessarily linear forward operators. The method builds on ideas from classical regularization theory and recent advances in deep learning to perform learning while making use of prior information about the inverse problem encoded in the forward operator, noise model and a regularizing functional. The method results in a gradient-like iterative scheme, where the "gradient" component is learned using a convolutional network that includes the gradients of the data discrepancy and regularizer as input in each iteration. We present results of such a partially learned gradient scheme on a non-linear tomographic inversion problem with simulated data from both the Sheep-Logan phantom as well as a head CT. The outcome is compared against FBP and TV reconstruction and the proposed method provides a 5.4 dB PSNR improvement over the TV reconstruction while being significantly faster, giving reconstructions of 512 x 512 volumes in about 0.4 seconds using a single GPU

Compressed Sensing Based Reconstruction Algorithm for X-ray Dose Reduction in Synchrotron Source Micro Computed Tomography

Synchrotron computed tomography requires a large number of angular projections to reconstruct tomographic images with high resolution for detailed and accurate diagnosis. However, this exposes the specimen to a large amount of x-ray radiation. Furthermore, this increases scan time and, consequently, the likelihood of involuntary specimen movements. One approach for decreasing the total scan time and radiation dose is to reduce the number of projection views needed to reconstruct the images. However, the aliasing artifacts appearing in the image due to the reduced number of projection data, visibly degrade the image quality. According to the compressed sensing theory, a signal can be accurately reconstructed from highly undersampled data by solving an optimization problem, provided that the signal can be sparsely represented in a predefined transform domain. Therefore, this thesis is mainly concerned with designing compressed sensing-based reconstruction algorithms to suppress aliasing artifacts while preserving spatial resolution in the resulting reconstructed image. First, the reduced-view synchrotron computed tomography reconstruction is formulated as a total variation regularized compressed sensing problem. The Douglas-Rachford Splitting and the randomized Kaczmarz methods are utilized to solve the optimization problem of the compressed sensing formulation. In contrast with the first part, where consistent simulated projection data are generated for image reconstruction, the reduced-view inconsistent real ex-vivo synchrotron absorption contrast micro computed tomography bone data are used in the second part. A gradient regularized compressed sensing problem is formulated, and the Douglas-Rachford Splitting and the preconditioned conjugate gradient methods are utilized to solve the optimization problem of the compressed sensing formulation. The wavelet image denoising algorithm is used as the post-processing algorithm to attenuate the unwanted staircase artifact generated by the reconstruction algorithm. Finally, a noisy and highly reduced-view inconsistent real in-vivo synchrotron phase-contrast computed tomography bone data are used for image reconstruction. A combination of prior image constrained compressed sensing framework, and the wavelet regularization is formulated, and the Douglas-Rachford Splitting and the preconditioned conjugate gradient methods are utilized to solve the optimization problem of the compressed sensing formulation. The prior image constrained compressed sensing framework takes advantage of the prior image to promote the sparsity of the target image. It may lead to an unwanted staircase artifact when applied to noisy and texture images, so the wavelet regularization is used to attenuate the unwanted staircase artifact generated by the prior image constrained compressed sensing reconstruction algorithm. The visual and quantitative performance assessments with the reduced-view simulated and real computed tomography data from canine prostate tissue, rat forelimb, and femoral cortical bone samples, show that the proposed algorithms have fewer artifacts and reconstruction errors than other conventional reconstruction algorithms at the same x-ray dose

Projected Newton Method for noise constrained Tikhonov regularization

Tikhonov regularization is a popular approach to obtain a meaningful solution for ill-conditioned linear least squares problems. A relatively simple way of choosing a good regularization parameter is given by Morozov's discrepancy principle. However, most approaches require the solution of the Tikhonov problem for many different values of the regularization parameter, which is computationally demanding for large scale problems. We propose a new and efficient algorithm which simultaneously solves the Tikhonov problem and finds the corresponding regularization parameter such that the discrepancy principle is satisfied. We achieve this by formulating the problem as a nonlinear system of equations and solving this system using a line search method. We obtain a good search direction by projecting the problem onto a low dimensional Krylov subspace and computing the Newton direction for the projected problem. This projected Newton direction, which is significantly less computationally expensive to calculate than the true Newton direction, is then combined with a backtracking line search to obtain a globally convergent algorithm, which we refer to as the Projected Newton method. We prove convergence of the algorithm and illustrate the improved performance over current state-of-the-art solvers with some numerical experiments

해부학적 유도 PET 재구성: 매끄럽지 않은 사전 함수부터 딥러닝 접근까지

학위논문 (박사) -- 서울대학교 대학원 : 의과대학 의과학과, 2021. 2. 이재성.Advances in simultaneous positron emission tomography/magnetic resonance imaging (PET/MRI) technology have led to an active investigation of the anatomy-guided regularized PET image reconstruction algorithm based on MR images. Among the various priors proposed for anatomy-guided regularized PET image reconstruction, Bowsher’s method based on second-order smoothing priors sometimes suffers from over-smoothing of detailed structures. Therefore, in this study, we propose a Bowsher prior based on the l1 norm and an iteratively reweighting scheme to overcome the limitation of the original Bowsher method. In addition, we have derived a closed solution for iterative image reconstruction based on this non-smooth prior. A comparison study between the original l2 and proposed l1 Bowsher priors were conducted using computer simulation and real human data. In the simulation and real data application, small lesions with abnormal PET uptake were better detected by the proposed l1 Bowsher prior methods than the original Bowsher prior. The original l2 Bowsher leads to a decreased PET intensity in small lesions when there is no clear separation between the lesions and surrounding tissue in the anatomical prior. However, the proposed l1 Bowsher prior methods showed better contrast between the tumors and surrounding tissues owing to the intrinsic edge-preserving property of the prior which is attributed to the sparseness induced by l1 norm, especially in the iterative reweighting scheme. Besides, the proposed methods demonstrated lower bias and less hyper-parameter dependency on PET intensity estimation in the regions with matched anatomical boundaries in PET and MRI. Moreover, based on the formulation of l1 Bowsher prior, the unrolled network containing the conventional maximum-likelihood expectation-maximization (ML-EM) module was also proposed. The convolutional layers successfully learned the distribution of anatomically-guided PET images and the EM module corrected the intermediate outputs by comparing them with sinograms. The proposed unrolled network showed better performance than ordinary U-Net, where the regional uptake is less biased and deviated. Therefore, these methods will help improve the PET image quality based on the anatomical side information.양전자방출단층촬영 / 자기공명영상 (PET/MRI) 동시 획득 기술의 발전으로 MR 영상을 기반으로 한 해부학적 사전 함수로 정규화 된 PET 영상 재구성 알고리즘에 대한 심도있는 평가가 이루어졌다. 해부학 기반으로 정규화 된 PET 이미지 재구성을 위해 제안 된 다양한 사전 중 2차 평활화 사전함수에 기반한 Bowsher의 방법은 때때로 세부 구조의 과도한 평활화로 어려움을 겪는다. 따라서 본 연구에서는 원래 Bowsher 방법의 한계를 극복하기 위해 l1 norm에 기반한 Bowsher 사전 함수와 반복적인 재가중치 기법을 제안한다. 또한, 우리는 이 매끄럽지 않은 사전 함수를 이용한 반복적 이미지 재구성에 대해 닫힌 해를 도출했다. 원래 l2와 제안 된 l1 Bowsher 사전 함수 간의 비교 연구는 컴퓨터 시뮬레이션과 실제 데이터를 사용하여 수행되었다. 시뮬레이션 및 실제 데이터에서 비정상적인 PET 흡수를 가진 작은 병변은 원래 Bowsher 이전보다 제안 된 l1 Bowsher 사전 방법으로 더 잘 감지되었다. 원래의 l2 Bowsher는 해부학적 영상에서 병변과 주변 조직 사이에 명확한 분리가 없을 때 작은 병변에서의 PET 강도를 감소시킨다. 그러나 제안 된 l1 Bowsher 사전 방법은 특히 반복적 재가중치 기법에서 l1 노름에 의해 유도된 희소성에 기인한 특성으로 인해 종양과 주변 조직 사이에 더 나은 대비를 보여주었다. 또한 제안된 방법은 PET과 MRI의 해부학적 경계가 일치하는 영역에서 PET 강도 추정에 대한 편향이 더 낮고 하이퍼 파라미터 종속성이 적음을 보여주었다. 또한, l1Bowsher 사전 함수의 닫힌 해를 기반으로 기존의 ML-EM (maximum-likelihood expectation-maximization) 모듈을 포함하는 펼쳐진 네트워크도 제안되었다. 컨볼루션 레이어는 해부학적으로 유도 재구성된 PET 이미지의 분포를 성공적으로 학습했으며, EM 모듈은 중간 출력들을 사이노그램과 비교하여 결과 이미지가 잘 들어맞게 수정했다. 제안된 펼쳐진 네트워크는 지역의 흡수선량이 덜 편향되고 편차가 적어, 일반 U-Net보다 더 나은 성능을 보여주었다. 따라서 이러한 방법들은 해부학적 정보를 기반으로 PET 이미지 품질을 향상시키는 데 유용할 것이다.Chapter 1. Introduction 1 1.1. Backgrounds 1 1.1.1. Positron Emission Tomography 1 1.1.2. Maximum a Posterior Reconstruction 1 1.1.3. Anatomical Prior 2 1.1.4. Proposed l_1 Bowsher Prior 3 1.1.5. Deep Learning for MR-less Application 4 1.2. Purpose of the Research 4 Chapter 2. Anatomically-guided PET Reconstruction Using Bowsher Prior 6 2.1. Backgrounds 6 2.1.1. PET Data Model 6 2.1.2. Original Bowsher Prior 7 2.2. Methods and Materials 8 2.2.1. Proposed l_1 Bowsher Prior 8 2.2.2. Iterative Reweighting 13 2.2.3. Computer Simulations 15 2.2.4. Human Data 16 2.2.5. Image Analysis 17 2.3. Results 19 2.3.1. Simulation with Brain Phantom 19 2.3.2.Human Data 20 2.4. Discussions 25 Chapter 3. Deep Learning Approach for Anatomically-guided PET Reconstruction 31 3.1. Backgrounds 31 3.2. Methods and Materials 33 3.2.1. Douglas-Rachford Splitting 33 3.2.2. Network Architecture 34 3.2.3. Dataset and Training Details 35 3.2.4. Image Analysis 36 3.3. Results 37 3.4. Discussions 38 Chapter 4. Conclusions 40 Bibliography 41 Abstract in Korean (국문 초록) 52Docto

Iterative CT reconstruction from few projections for the nondestructive post irradiation examination of nuclear fuel assemblies

The core components (e.g. fuel assemblies, spacer grids, control rods) of the nuclear reactors encounter harsh environment due to high temperature, physical stress, and a tremendous level of radiation. The integrity of these elements is crucial for safe operation of the nuclear power plants. The Post Irradiation Examination (PIE) can reveal information about the integrity of the elements during normal operations and off‐normal events. Computed tomography (CT) is a tool for evaluating the structural integrity of elements non-destructively. CT requires many projections to be acquired from different view angles after which a mathematical algorithm is adopted for reconstruction. Obtaining many projections is laborious and expensive in nuclear industries. Reconstructions from a small number of projections are explored to achieve faster and cost-efficient PIE. Classical reconstruction algorithms (e.g. filtered back projection) cannot offer stable reconstructions from few projections and create severe streaking artifacts. In this thesis, conventional algorithms are reviewed, and new algorithms are developed for reconstructions of the nuclear fuel assemblies using few projections. CT reconstruction from few projections falls into two categories: the sparse-view CT and the limited-angle CT or tomosynthesis. Iterative reconstruction algorithms are developed for both cases in the field of compressed sensing (CS). The performance of the algorithms is assessed using simulated projections and validated through real projections. The thesis also describes the systematic strategy towards establishing the conditions of reconstructions and finds the optimal imaging parameters for reconstructions of the fuel assemblies from few projections. --Abstract, page iii